NBALabAssignment_6
Josephine Johannes
10/12/2021
High Performing Players
Rationale for Variable Selection
To find the best players, it was crucial that the variables taken into account consisted of the number of games that the players started in and the total number of points that a player had for the season. The number of games that a player has started is important because it shows how other teams thought highly of the player. The field goals percentage is also important because it has the overall number of field goals including 3-pointers and 2-pointers that the players correctly executed. The amount of total rebounds is also essential to know because it can be an important factor of determining possession of the ball in future games. The number of assists is crtiical as a variable because it shows how the player is the second-to-last player to touch the ball before a point is scored. Lastly, the number of points that a player had for the season is important because it shows how many times the player was able to execute a play and the throw perfectly.
Details of the Approach
This study will determine which professional players are a high-performance athlete with a low salary. To ensure that this determination is accurate, the data will first be normalized. Then, the variables will be selected and run through the k-means algorithm. The number of clusters will be determined by the Nbclust package. Then the data will be plotted to visualize the data and then the results will be validated.
####Merging the Datasets
# converting the data into data frame format
nba <- as.data.frame(nba)
nba_sal <-as.data.frame(nba_sal)
nba <- merge(nba, nba_sal)
# to remove multiple players: nba <- distinct_at(nba, Player)
nba <- nba[complete.cases(nba), ]
nba <- as.data.frame(nba)
# normalize the columns before they're added
nba$GS <- normalize(nba$GS)
nba$FG <- scale(nba$FG, center= TRUE, scale = TRUE)
nba$TRB <- normalize(nba$TRB)
nba$AST <- normalize(nba$AST)
nba$PTS <- normalize(nba$PTS)
# Subsetting the data with the selected variables
clust_data = nba[, c("Age", "G", "GS","MP", "FG","3P%","2P%","eFG%", "FT%", "TRB", "AST","STL", "BLK", "TOV", "PTS" )] #
clust_data = normalize(clust_data)
View(clust_data)
View(nba)Finding the ideal number of clusters
# using the NbClust algorithm to find the ideal number of clusters
(nbclust_obj_nba = NbClust(data = clust_data, method= "kmeans"))
## *** : The Hubert index is a graphical method of determining the number of clusters.
## In the plot of Hubert index, we seek a significant knee that corresponds to a
## significant increase of the value of the measure i.e the significant peak in Hubert
## index second differences plot.
## 
## *** : The D index is a graphical method of determining the number of clusters.
## In the plot of D index, we seek a significant knee (the significant peak in Dindex
## second differences plot) that corresponds to a significant increase of the value of
## the measure.
##
## *******************************************************************
## * Among all indices:
## * 4 proposed 2 as the best number of clusters
## * 13 proposed 3 as the best number of clusters
## * 1 proposed 4 as the best number of clusters
## * 1 proposed 9 as the best number of clusters
## * 1 proposed 14 as the best number of clusters
## * 3 proposed 15 as the best number of clusters
##
## ***** Conclusion *****
##
## * According to the majority rule, the best number of clusters is 3
##
##
## *******************************************************************## $All.index
## KL CH Hartigan CCC Scott Marriot TrCovW TraceW Friedman
## 2 3.8170 1028.282 598.7915 25.1521 4096.224 0 4.1012 8.0486 8934.998
## 3 3.6694 1549.858 255.9445 -11.0346 4556.171 0 0.6614 3.2994 9030.297
## 4 1.4964 1751.556 264.5820 -20.7778 4841.865 0 0.2430 2.0408 9348.392
## 5 2.4717 2213.999 143.1475 -21.5722 5116.271 0 0.0842 1.2451 9374.905
## 6 1.6078 2407.889 110.7261 -26.1628 5275.322 0 0.0437 0.9246 9317.825
## 7 1.4584 2558.077 91.0065 -33.1561 5420.672 0 0.0253 0.7288 9423.771
## 8 0.8839 2684.600 128.1827 -27.8202 5546.898 0 0.0159 0.5966 9680.998
## 9 3.7590 3091.889 40.7452 -19.7448 5689.576 0 0.0081 0.4545 9812.636
## 10 0.9499 3019.267 45.4503 -17.8042 5769.981 0 0.0064 0.4134 9821.334
## 11 1.1697 3017.178 42.1336 -15.3673 5848.723 0 0.0049 0.3719 9913.053
## 12 2.1757 3023.224 22.2266 -13.0847 5913.528 0 0.0038 0.3370 9998.356
## 13 0.4059 2917.651 53.1801 -12.3230 5963.893 0 0.0033 0.3195 10089.425
## 14 1.1128 3043.418 53.1041 -9.0067 6094.934 0 0.0023 0.2825 10370.676
## 15 2.0227 3193.369 29.3662 -5.6263 6178.730 0 0.0016 0.2496 10484.558
## Rubin Cindex DB Silhouette Duda Pseudot2 Beale Ratkowsky Ball
## 2 13.7424 0.2866 0.6343 0.5821 0.4927 282.1426 10.5849 0.3475 4.0243
## 3 33.5232 0.2679 0.5850 0.5840 0.9395 14.1150 0.6614 0.3253 1.0998
## 4 54.1981 0.2874 0.6124 0.5435 0.7429 62.9932 3.5441 0.2967 0.5102
## 5 88.8353 0.2381 0.5945 0.5497 1.0938 -13.1264 -0.8775 0.2729 0.2490
## 6 119.6260 0.2678 0.6264 0.5219 1.4670 -32.1524 -3.2482 0.2520 0.1541
## 7 151.7759 0.2473 0.6553 0.4963 1.2757 -19.8815 -2.1941 0.2325 0.1041
## 8 185.3831 0.2922 0.6684 0.4913 1.0295 -2.4926 -0.2902 0.2191 0.0746
## 9 243.3414 0.2554 0.6655 0.4969 2.4989 -69.5787 -6.0508 0.2084 0.0505
## 10 267.5835 0.2450 0.7039 0.4790 0.6096 41.6194 6.4905 0.1993 0.0413
## 11 297.3917 0.2339 0.7193 0.4660 4.2484 -71.1093 -7.6713 0.1904 0.0338
## 12 328.1784 0.2233 0.7357 0.4450 2.5733 -39.1293 -6.1287 0.1824 0.0281
## 13 346.1447 0.2204 0.7492 0.4363 0.7197 18.3067 3.9196 0.1761 0.0246
## 14 391.5965 0.2309 0.7848 0.4306 2.7733 -30.0524 -6.3167 0.1711 0.0202
## 15 443.0702 0.2153 0.7940 0.4210 2.4788 -29.8292 -5.8936 0.1650 0.0166
## Ptbiserial Frey McClain Dunn Hubert SDindex Dindex SDbw
## 2 0.6554 0.6636 0.3727 0.0210 0.0357 28.7353 0.1210 1.5371
## 3 0.6628 1.1713 0.5186 0.0264 0.0456 17.6744 0.0774 0.5443
## 4 0.6048 0.8210 0.6503 0.0386 0.0499 17.8616 0.0622 0.5236
## 5 0.5627 1.1208 0.7190 0.0314 0.0533 18.8373 0.0480 0.2718
## 6 0.5251 1.2721 0.7925 0.0406 0.0547 22.0371 0.0421 0.3480
## 7 0.4879 0.8201 0.8801 0.0370 0.0554 27.6344 0.0375 0.3473
## 8 0.4714 0.8812 0.9031 0.0365 0.0562 28.9251 0.0339 0.1567
## 9 0.4437 1.5847 0.9468 0.0552 0.0566 33.9304 0.0296 0.0582
## 10 0.4274 1.7697 0.9995 0.0552 0.0567 49.2226 0.0281 0.0299
## 11 0.4064 1.5159 1.0798 0.0365 0.0568 56.8172 0.0265 0.0310
## 12 0.3885 2.7851 1.1518 0.0578 0.0569 62.7915 0.0252 0.0268
## 13 0.3785 0.6694 1.2061 0.0584 0.0570 79.5621 0.0243 0.0236
## 14 0.3685 1.0415 1.2201 0.0651 0.0572 77.2647 0.0230 0.0214
## 15 0.3503 1.4111 1.2891 0.0514 0.0575 85.5277 0.0216 0.0155
##
## $All.CriticalValues
## CritValue_Duda CritValue_PseudoT2 Fvalue_Beale
## 2 0.8799 37.4091 0.0000
## 3 0.8666 33.7209 0.8243
## 4 0.8529 31.3800 0.0000
## 5 0.8461 27.8256 1.0000
## 6 0.8336 20.1670 1.0000
## 7 0.8108 21.4624 1.0000
## 8 0.8015 21.5528 1.0000
## 9 0.7862 31.5395 1.0000
## 10 0.8043 15.8138 0.0000
## 11 0.7681 28.0705 1.0000
## 12 0.7655 19.6097 1.0000
## 13 0.7779 13.4216 0.0000
## 14 0.7256 17.7759 1.0000
## 15 0.7256 18.9105 1.0000
##
## $Best.nc
## KL CH Hartigan CCC Scott Marriot TrCovW TraceW
## Number_clusters 2.000 15.000 3.000 2.0000 3.0000 3 3.0000 3.0000
## Value_Index 3.817 3193.369 342.847 25.1521 459.9466 0 3.4398 3.4906
## Friedman Rubin Cindex DB Silhouette Duda PseudoT2
## Number_clusters 4.000 9.0000 15.0000 3.000 3.000 3.0000 3.000
## Value_Index 318.096 -33.7163 0.2153 0.585 0.584 0.9395 14.115
## Beale Ratkowsky Ball PtBiserial Frey McClain Dunn Hubert
## Number_clusters 3.0000 2.0000 3.0000 3.0000 1 2.0000 14.0000 0
## Value_Index 0.6614 0.3475 2.9245 0.6628 NA 0.3727 0.0651 0
## SDindex Dindex SDbw
## Number_clusters 3.0000 0 15.0000
## Value_Index 17.6744 0 0.0155
##
## $Best.partition
## 1 2 3 4 5 6 7 9 10 11 12 13 14 15 16 18 19 20 21 22
## 1 1 2 2 2 1 1 1 2 2 2 2 2 2 2 1 1 2 3 1
## 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
## 1 3 2 1 1 2 3 2 3 1 1 1 2 1 3 1 2 3 3 1
## 43 44 45 46 47 48 49 50 51 52 53 54 55 57 58 59 60 61 62 63
## 2 2 1 3 2 1 1 2 2 3 2 3 2 3 3 2 1 1 2 1
## 65 66 67 68 69 70 72 73 74 75 76 77 79 80 81 83 84 85 86 87
## 3 2 1 1 3 1 1 1 2 1 2 2 3 1 1 2 1 1 3 2
## 88 89 90 91 92 93 94 95 96 97 98 99 100 101 103 104 105 106 107 108
## 1 2 2 3 3 2 1 3 1 2 2 1 3 1 3 3 3 1 1 3
## 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
## 2 2 2 1 1 3 2 1 2 2 1 3 1 1 3 3 3 3 1 3
## 129 131 132 133 134 135 136 138 139 140 141 143 144 145 146 147 149 150 151 152
## 3 3 3 1 2 1 2 2 3 3 2 3 1 1 1 1 1 2 3 1
## 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 173 174
## 3 1 1 3 2 1 3 2 2 1 3 2 1 1 1 3 2 2 1 2
## 175 177 178 179 180 181 182 183 185 186 187 188 190 191 193 194 195 196 197 198
## 3 2 1 1 2 1 1 1 1 3 3 2 2 2 1 2 2 2 3 1
## 199 200 201 202 203 204 206 208 209 210 211 212 214 215 216 217 218 219 220 221
## 3 3 2 1 1 1 1 1 3 2 2 2 3 2 3 3 1 3 1 2
## 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241
## 2 1 3 3 3 3 2 1 1 3 2 2 2 3 1 1 1 1 2 3
## 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261
## 3 2 2 2 1 3 2 3 3 3 1 3 1 1 3 2 1 3 2 2
## 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281
## 2 1 3 2 2 1 1 3 3 3 2 1 3 1 2 1 1 3 1 3
## 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301
## 1 3 2 3 1 2 2 3 3 1 1 1 1 2 1 2 3 3 1 2
## 302 303 305 306 307 308 309 310 311 312 314 315 316 317 318 319 320 321 322 323
## 2 1 2 1 1 3 3 1 2 3 2 2 3 3 2 2 3 2 1 2
## 324 326 327 328 329 330 331 332 334 338 339 340 341 342 343 344 345 346 348 349
## 1 1 2 3 3 2 3 3 3 2 1 2 2 3 3 3 1 1 2 2
## 350 351 352 353 355 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371
## 3 1 3 1 1 2 2 2 2 1 3 1 3 3 3 3 1 2 1 2
## 372 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392
## 2 1 2 3 1 3 1 3 2 1 2 1 1 3 3 3 1 2 2 1
## 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 410 411 412 413
## 2 1 3 1 3 1 1 2 2 2 1 2 2 1 1 1 3 3 1 1
## 414 415 416 417 418 419 420 421 423 424 425 426 427 428 429 431 432 433 434 435
## 2 2 2 3 1 3 1 2 2 3 2 1 2 1 2 1 2 2 3 1
## 436 438 439 440 441 443 444 445 446 448 449 450 451 452 453 454 455 456
## 1 2 1 1 2 2 1 1 1 2 1 3 1 2 1 3 2 3# subset the first row from Best.nc and convert to a data frame
freq_k_nba = nbclust_obj_nba$Best.nc[1,]
freq_k_nba = data.frame(freq_k_nba)
#Plot the recommended number of clusters as a histogram
ggplot(freq_k_nba, aes(x = freq_k_nba)) + geom_bar()+ scale_x_continuous(breaks = seq(0, 15, by = 1)) + scale_y_continuous(breaks = seq(0, 12, by = 1)) + labs(x = "Number of Clusters", y = "Number of Votes", title = "Cluster Analysis")
From the cluster analysis, the recommended number of clusters is 3.
Run the K-means algorithm
# Run an algorithm with centers and make the results reproducible with set.seed
set.seed(1)
kmeans_obj_nba = kmeans(clust_data, centers = 3, algorithm = "Lloyd")
head(kmeans_obj_nba)## $cluster
## 1 2 3 4 5 6 7 9 10 11 12 13 14 15 16 18 19 20 21 22
## 1 1 2 2 2 1 1 1 2 2 2 2 2 2 2 1 1 2 3 1
## 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
## 1 3 2 1 1 2 3 2 3 1 1 1 2 1 3 1 2 3 3 1
## 43 44 45 46 47 48 49 50 51 52 53 54 55 57 58 59 60 61 62 63
## 2 2 1 3 2 1 1 2 2 3 2 3 2 3 3 2 1 1 2 1
## 65 66 67 68 69 70 72 73 74 75 76 77 79 80 81 83 84 85 86 87
## 3 2 2 1 3 1 1 1 2 1 2 2 3 1 1 2 1 1 3 2
## 88 89 90 91 92 93 94 95 96 97 98 99 100 101 103 104 105 106 107 108
## 1 2 2 3 3 2 1 3 1 2 2 1 3 1 3 3 3 1 1 3
## 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
## 2 2 2 1 1 1 2 1 2 2 1 3 1 1 3 3 3 3 1 3
## 129 131 132 133 134 135 136 138 139 140 141 143 144 145 146 147 149 150 151 152
## 3 3 3 1 2 2 2 2 3 3 2 3 1 1 1 1 1 2 3 1
## 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 173 174
## 3 1 1 3 2 1 3 2 2 1 3 2 1 1 1 3 2 2 1 2
## 175 177 178 179 180 181 182 183 185 186 187 188 190 191 193 194 195 196 197 198
## 3 2 1 1 2 1 1 1 1 3 3 2 2 2 1 2 2 2 3 1
## 199 200 201 202 203 204 206 208 209 210 211 212 214 215 216 217 218 219 220 221
## 3 3 2 1 1 1 1 1 3 2 2 2 3 2 3 3 1 3 1 2
## 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241
## 2 1 3 3 3 3 2 1 1 3 2 2 2 3 1 1 1 1 2 3
## 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261
## 3 2 2 2 1 3 2 3 3 3 1 1 1 1 3 2 1 3 2 2
## 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281
## 2 1 3 2 2 1 1 3 3 3 2 1 3 1 2 1 1 3 1 3
## 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301
## 1 3 2 3 1 2 2 3 3 1 1 1 1 2 1 2 3 3 1 2
## 302 303 305 306 307 308 309 310 311 312 314 315 316 317 318 319 320 321 322 323
## 2 1 2 1 1 3 3 1 2 3 2 2 3 3 2 2 3 2 1 2
## 324 326 327 328 329 330 331 332 334 338 339 340 341 342 343 344 345 346 348 349
## 1 1 2 3 3 2 3 3 3 2 1 2 2 3 3 3 1 1 2 2
## 350 351 352 353 355 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371
## 3 1 3 1 1 2 2 2 2 1 1 1 3 3 3 3 1 2 1 2
## 372 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392
## 2 1 2 3 1 3 1 3 2 1 2 1 1 3 1 3 1 2 2 1
## 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 410 411 412 413
## 2 1 3 1 3 1 1 2 2 2 1 2 2 1 1 1 3 3 1 1
## 414 415 416 417 418 419 420 421 423 424 425 426 427 428 429 431 432 433 434 435
## 2 2 2 3 1 3 1 2 2 3 2 1 2 1 2 1 2 2 3 1
## 436 438 439 440 441 443 444 445 446 448 449 450 451 452 453 454 455 456
## 1 2 1 1 2 2 1 1 1 2 1 3 1 2 1 3 2 3
##
## $centers
## Age G GS MP FG 3P%
## 1 0.02054189 0.02174322 0.0011414986 0.4705292 0.0008859540 0.001169983
## 2 0.01942502 0.01240011 0.0009604542 0.1514115 0.0002508212 0.001138877
## 3 0.02041617 0.02546459 0.0014791082 0.7663710 0.0017975619 0.001182005
## 2P% eFG% FT% TRB AST STL
## 1 0.001297632 0.001314167 0.001486702 0.0010937955 0.0010438441 0.016220821
## 2 0.001293589 0.001293742 0.001436110 0.0009769641 0.0009562825 0.005666418
## 3 0.001310994 0.001323228 0.001507254 0.0012125384 0.0011898356 0.024367383
## BLK TOV PTS
## 1 0.010484227 0.024616176 0.001107860
## 2 0.004521996 0.008578561 0.000972334
## 3 0.014936477 0.049013315 0.001304945
##
## $totss
## [1] 27.9435
##
## $withinss
## [1] 1.133004 1.117633 1.051747
##
## $tot.withinss
## [1] 3.302384
##
## $betweenss
## [1] 24.64111clusters = as.factor(kmeans_obj_nba$cluster)
clusters = as.data.frame(clusters)Visualizing Various Plots and Correlations
####Correlation between the Salary and Points
# This plot shows the correlation between the games started and the points each player has
sal_clusters = as.factor(kmeans_obj_nba$cluster)
b <-ggplot(nba, aes(x = Salary, y = PTS, color = sal_clusters, text=Player))+geom_point(size = 6)+ggtitle("Salary vs. Points for NBA Basketball players") +xlab("Salary")+ylab("Number of Points")+scale_shape_manual(name = "Cluster", labels = c("Cluster 1", "Cluster 2", "Cluster 3"), values = c("1", "2", "3"))+ theme_light()
ggplotly(b, tooltip="text")####Correlation between Salary and Games Started
sal_clusters = as.factor(kmeans_obj_nba$cluster)
c <-ggplot(nba, aes(x = Salary, y = GS, color = sal_clusters, text=Player))+geom_point(size = 6) + ggtitle("Salary vs. Games started by NBA Basketball players") + xlab("Salary")+ylab("Number of Games Started") + scale_shape_manual(name = "Cluster", labels = c("Cluster 1", "Cluster 2", "Cluster 3"), values = c("1", "2","3"))+ theme_light()
ggplotly(c, tooltip="text")####Correlation between the Salary and Field Goals
sal_clusters = as.factor(kmeans_obj_nba$cluster)
a <-ggplot(nba, aes(x = Salary, y = FG, color = sal_clusters, text = Player))+geom_point(size = 6) + ggtitle("Salary vs. Field Goals for NBA Basketball players") + xlab("Salary")+ylab("Number of Field Goals") + scale_shape_manual(name = "Cluster", labels = c("Cluster 1", "Cluster 2", "Cluster 3"), values = c("1", "2", "3"))+ theme_light()
ggplotly(a, tooltip="text")####Correlation between the Salary and Number of Assists
sal_clusters = as.factor(kmeans_obj_nba$cluster)
a <-ggplot(nba, aes(x = Salary, y = AST, color = sal_clusters, text = Player))+geom_point(size = 6) + ggtitle("Salary vs. Assists for NBA Basketball players") + xlab("Salary")+ylab("Number of Assists") + scale_shape_manual(name = "Cluster", labels = c("Cluster 1", "Cluster 2", "Cluster 3"), values = c("1", "2", "3"))+ theme_light()
ggplotly(a, tooltip="text")Final Recommendations
I would recommend Trae Young, Luka Don, Donovan Mitchell, and Zion Williamson I recommend these three players because they have done well in terms of how many field goals they’ve completed during the last season as well as the number of points that they accumulated. They are also not paid as well as the other athletes and seem to be high-performing. These players seem to have a high number of points compared to others and are on the lower end of the salary spectrum.